This can be written in the form of a Ward-identity [1]}, [2]}, [3]}, [4]}, [5]}, [6]} as,
\(\boxed{{\beta , out} Q^{out}_S(f){\alpha , in} - {\beta , out} Q^{in}_S(f){\alpha , in} = \bigg ( \sum _{i\in \alpha } q_i f(x_i) - \sum _{i\in \beta } q_i f(x_i) \bigg ) {\beta , out}{\alpha , in}}\) 
